1992 AHSME Problems/Problem 11
Problem
The ratio of the radii of two concentric circles is . If is a diameter of the larger circle, is a chord of the larger circle that is tangent to the smaller circle, and , then the radius of the larger circle is
Solution (Similarity)
We are given that is tangent to the smaller circle. Using that, we know where the circle intersects , it creates a right triangle. We can also point out that since is the diameter of the bigger circle and triangle is inscribed the semi-circle, that angle is a right angle. Therefore, we have similar triangles. Let's label the center of the smaller circle (which is also the center of the larger circle) as . Let's also label the point where the smaller circle intersects as . So is similar to . Since is the radius of the smaller circle, call the length and since is the radius of the bigger circle, call that length . The diameter, is . So,
But they are asking for the larger circle radius, so
See also
1992 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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